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In the given fig, the line segments AB and CD intersect at a point M in such a way that AM = MD and CM = MB. Prove that, AC = BD but AC may not be parallel to BD.?
Most Upvoted Answer
In the given fig, the line segments AB and CD intersect at a point M i...
Proof:
Given: AB and CD are two line segments intersecting at M such that AM = MD and CM = MB.
To prove: AC = BD
Proof:
Let us draw a diagram for better understanding:
https://www.edurev.in/api/img/ncert-solutions/2024388.jpg" />
In ∆AMB and ∆DMC,
AM = MD (Given)
MB = MC (Given)
∠AMB = ∠DMC (Vertically opposite angles)
Therefore, ∆AMB ≅ ∆DMC (By SAS congruence rule)
So, AB = DC (By CPCT)
Now, in ∆AMC and ∆BMD,
AM = MD (Given)
MB = MC (Given)
∠AMC = ∠BMD (Vertically opposite angles)
Therefore, ∆AMC ≅ ∆BMD (By SAS congruence rule)
So, AC = BD (By CPCT)
Hence, we have proved that AC = BD.
Explanation:
The given figure is of two line segments AB and CD, intersecting at M. We need to prove that AC = BD.
We first draw a diagram of the given segment AB and CD intersecting at M. Then we join AM, MD, MB, and MC. Since AM = MD and CM = MB, we can say that ∆AMB ≅ ∆DMC by SAS congruence rule. Therefore, AB = DC (By CPCT).
Now, we join AC and BD. In ∆AMC and ∆BMD, we can see that AM = MD and MB = MC. Also, we can say that ∠AMC = ∠BMD since they are vertically opposite angles. Therefore, ∆AMC ≅ ∆BMD by SAS congruence rule. Hence, we can say that AC = BD (By CPCT).
Therefore, we have proved that AC = BD. It is important to note that AC may not be parallel to BD in this case.
Given: AB and CD are two line segments intersecting at M such that AM = MD and CM = MB.
To prove: AC = BD
Proof:
Let us draw a diagram for better understanding:
In ∆AMB and ∆DMC,
AM = MD (Given)
MB = MC (Given)
∠AMB = ∠DMC (Vertically opposite angles)
Therefore, ∆AMB ≅ ∆DMC (By SAS congruence rule)
So, AB = DC (By CPCT)
Now, in ∆AMC and ∆BMD,
AM = MD (Given)
MB = MC (Given)
∠AMC = ∠BMD (Vertically opposite angles)
Therefore, ∆AMC ≅ ∆BMD (By SAS congruence rule)
So, AC = BD (By CPCT)
Hence, we have proved that AC = BD.
Explanation:
The given figure is of two line segments AB and CD, intersecting at M. We need to prove that AC = BD.
We first draw a diagram of the given segment AB and CD intersecting at M. Then we join AM, MD, MB, and MC. Since AM = MD and CM = MB, we can say that ∆AMB ≅ ∆DMC by SAS congruence rule. Therefore, AB = DC (By CPCT).
Now, we join AC and BD. In ∆AMC and ∆BMD, we can see that AM = MD and MB = MC. Also, we can say that ∠AMC = ∠BMD since they are vertically opposite angles. Therefore, ∆AMC ≅ ∆BMD by SAS congruence rule. Hence, we can say that AC = BD (By CPCT).
Therefore, we have proved that AC = BD. It is important to note that AC may not be parallel to BD in this case.
Community Answer
In the given fig, the line segments AB and CD intersect at a point M i...
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In the given fig, the line segments AB and CD intersect at a point M in such a way that AM = MD and CM = MB. Prove that, AC = BD but AC may not be parallel to BD.?
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In the given fig, the line segments AB and CD intersect at a point M in such a way that AM = MD and CM = MB. Prove that, AC = BD but AC may not be parallel to BD.? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In the given fig, the line segments AB and CD intersect at a point M in such a way that AM = MD and CM = MB. Prove that, AC = BD but AC may not be parallel to BD.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the given fig, the line segments AB and CD intersect at a point M in such a way that AM = MD and CM = MB. Prove that, AC = BD but AC may not be parallel to BD.?.
In the given fig, the line segments AB and CD intersect at a point M in such a way that AM = MD and CM = MB. Prove that, AC = BD but AC may not be parallel to BD.? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In the given fig, the line segments AB and CD intersect at a point M in such a way that AM = MD and CM = MB. Prove that, AC = BD but AC may not be parallel to BD.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the given fig, the line segments AB and CD intersect at a point M in such a way that AM = MD and CM = MB. Prove that, AC = BD but AC may not be parallel to BD.?.
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